Hydrodynamic theory of quantum fluctuating superconductivity
Sean Hartnoll (Stanford)
Israel Institute for Advanced Study
Jerusalem — 2016
Based on: arXiv/1602.08171 [cond-mat.supr-con]
Collaborators
Richard Davison
(Harvard)
Luca Delacrétaz
(Stanford)
Blaise Goutéraux
(Stanford)
Hydrodynamic description of conventional metals •
Hydrodynamics:
→Universal low energy, long wavelength physics.
→Conserved charges, their currents, Goldstone bosons.
•
Conservation law:
•
@⇢ +r·j =0 @t Constitutive relation (derivative expansion):
j=
•
Dr⇢ + · · ·
Conductivity (Einstein relation): @⇢ j = Dr⇢ = D rµ = D E = E @µ
Comment on screening by Maxwell fields •
Charge in a metal does not diffuse, it decays exponentially.
•
This comes from solving Maxwell’s equations + Ohm’s law.
•
The Einstein relation for the conductivity still holds.
•
σ measured with respect to total, not external, electric field:
j = Etot =
Eext = ✏(!, k) 1
Eext = 1 i! (!, k) 2 k
i!D Eext 2 D(k + )
Superfluid hydrodynamics •
Phase ϕ of the order parameter appears in hydrodynamics.
•
1 u = r m
•
•
•
is the superfluid velocity.
‘Josephson relation’:
@ @u =r = rµ + · · · m @t @t Constitutive relation:
⇢s Dr⇢ + · · · j= u m (super-)Conductivity: ✓ ◆ ✓ ◆ ⇢s i ⇢s i j= + D rµ = + D E = (!)E m2 ! m2 !
Superconductivity •
∞ conductivity because: diffusion → second sound mode.
•
In a superconductor, the U(1) symmetry is gauged, i.e. coupled to electromagnetism.
•
This gaps out the Goldstone/sound mode in the same way the diffusive mode was previously gapped.
•
However, the conductivity is, as before, measured with respect to the total electric field. So the unscreened (superfluid) hydrodynamics determines the conductivities.
Vortices and supercurrent relaxation •
In two space dimensions, above picture incomplete.
•
Motion of vortices can wind and unwind the supercurrent.
•
Expect supercurrent
relaxation rate Ω:
⇢s (!) = 2 m
= 2⇡ 1 i! + ⌦
Vortices and supercurrent relaxation •
This problem is well understood in some regimes:
→ Thermal BKT proliferation of vortices above TBKT.
•
Classical picture: vortices pushed across sample by ‘superfluid Magnus force’
→ The core of the vortices is in the normal state.
→ Therefore, motion of vortices creates dissipation.
→ Get
nf Av ⌦⇠ [Bardeen-Stephen ’65] n
•
Much controversy, however, about whether (quantum) phase-disordered superconductors exist at T = 0.
In the remainder •
Lightening overview of some experiments.
•
Develop a fully quantum effective field theoretic formalism for the conductivity of phase-disordered superconductors.
•
Illustrate formalism with two examples:
(i) ‘Check’: Elegant (re)derivation of Bardeen-Stephen result.
(ii) Phase disordering by a Chern-Simons interaction
[‘topologically ordered superfluid vortex liquid’].
Superfluid-insulator transitions •
In two (spatial) dimensions, conventional theory suggests that as T→0 electrons will either localize or pair up.
•
That is, the phase of matter one expects to find is either an insulator or a superconductor.
•
Indeed, early experiments suggested that disordered thin films undergo superconductor-insulator transitions as a function of magnetic field or thickness (≈ 1/disorder).
Destroys superconductivity Favors localization
Superfluid-insulator transitions [Jaeger et al. ’89, Pb]
[Hebard and Paalanen ’90, α-InOx]
Metallic phases in two dimensions •
Problematically for ‘conventional’ understanding, in weakly disordered films a metallic phase intervenes
(at T = 0!) between the superconductor and insulator.
[Mason, Kapitulnik ’99, α-MoGe]
Metallic phases in two dimensions •
Often, the residual resistivity of the metallic phase is much smaller than the “normal state” resistivity of the material at temperatures above the “mean field” superconducting temperature.
•
Suggests the low energy degrees of freedom of the metallic phases are not the normal state quasiparticles.
•
Natural to think of as “failed superconductors” where (quantum!) phase fluctuations have destroyed phase coherence.
Metallic phases in two dimensions •
Direct motivation for our work: observation of a Drudelike peak in the metallic phase of InOx.
[Liu, Pan, Wen, Kim, Sambandamurthy, Armitage ’13]
Metallic phases in two dimensions •
The width of the Drude-like peak goes to zero at the same magnetic field where superconductivity appears.
[Liu, Pan, Wen, Kim, Sambandamurthy, Armitage ’13]
Memory matrix formalism •
Most discussions of this physics have involved semimicroscopic models with uncontrolled approximations.
•
Instead: work in a limit where a hierarchy of timescales allows an effective field theoretic approach.
•
Small parameter will be the supercurrent relaxation rate. I.e. want Ω ≪ T, etc.
•
(Approach inspired by studies in holographic systems over past few years, where slow mode was momentum.)
[Logic goes back to:
Götze and Wölfle ’72,
Forster ’75, …]
Memory matrix formalism •
Suppose that H = H0 + ε ΔH, with [ΔH,Jϕ] ≠ 0.
•
Then the decay of Jϕ is slow and dominates σ:
(!) =
2 JJ J J
•
1 + ··· i! + ⌦
But now we have a formula for Ω! : ⌦=✏
2
1 J J
lim
!!0
R Im Gi[ H,J ] i[ H,J ] (!)
!
. ✏=0
Spectral density of states into which
JΦ can decay. Cf. Fermi Golden rule.
Supercurrent relaxation Recap: if an ‘almost conserved’ operator carries current, rate of the decay determines the conductivity.
•
1 J = m
Z
•
In our case of interest today:
•
Need an interaction that doesn’t commute with JΦ.
•
Natural building block:
@f = ⇡ = @˙
d2 xr
@f = ⇢. @µ
i.e. charge density is canonically conjugate to the phase: [ (x), ⇢(y)] = i (x
y) .
Supercurrent relaxation •
•
•
Thus a simple, generic perturbation of the superfluid state is the short range Coulombic interaction:
Z
2 2 d x ⇢(x) . H= 2
At first glance looks like commutator is trivial total derivative:
Z
d2 xr⇢(x) i[ H, J ] = m However, the phase appearing in Jϕ is only defined outside of vortex cores! Above integral is then also only over the outside of vortex cores. Integral over all space vanishes:
→integral over vortex cores.
Supercurrent relaxation •
The memory matrix formula for Ω becomes an integral of the two point function of ρ over the vortex core.
•
Using the diffusive behavior of ρ in normal state, the Bardeen-Stephen formula drops out exactly.
⌦⇠
•
nf Av n
So we discover the quantum origin of this formula. Charge interactions enhance phase fluctuations: ⇢
&~
Supercurrent relaxation without parity •
With parity and time-reversal broken, a second very natural ΔH exists.
•
Suppose the low energy effective theory is coupled to an emergent Chern-Simons gauge field:
µ
µ
L = Lmatter + jµ (A + a ) •
1 µ⌫⇢ ✏ aµ @ ⌫ a⇢ 0 2
Integrating out the gauge field generates 0
µ⌫⇢
✏ @⇢ 0 L = jµ j⌫ 2 @ @
)
H=
0
2
Z
d2 k ⇢ (2⇡)2
z (r ⇥ j) k k + h.c. 2 k
Supercurrent relaxation without parity Non-locality of induced interaction leads to a nonzero time dependence of Jϕ everywhere. In fact:
•
i
i[ H, J ] =
0
m
ij
j
✏ J .
•
Rough physical picture:
Current = Flow of charge
→ Flow of emergent magnetic flux (CS term)
→ Flow of vortices
→ Relaxation of supercurrent in transverse direction!
•
Ω depends on charge flow in normal component.
Supercurrent relaxation without parity Result for conductivities:
•
xx
=
xy
•
=
m2 !(!⌦ + i(⌦2 + ⌦2H )) , 2 02 ⇢ 2 s ( i! + ⌦) + ⌦H 1 0
m2 ! 2 ⌦H , 02 ⇢ ( i! + ⌦)2 + ⌦2 s H
Feature: ‘supercyclotron resonance’ at H
!? = ±⌦
0
⇢s i⌦ = m2 ±1
1 0 (±
H 0
i
0)
.
Conductivities of the normal component
of superfluid.
Recap
[see arXiv/1602.08171]
•
Superfluid relaxation occurs if perturbations of effective Hamiltonian do not commute with the supercurrent.
•
Starting with perturbations of superfluid hydrodynamics gives controlled entry point. This works even if the underlying microscopic dynamics is strongly correlated.
•
Gave two examples, with and without parity:
(1) With parity: recovered Bardeen-Stephen.
(2) Without parity: ‘supercyclotron resonance’ determined by conductivities of normal component.